The correlated random walk is a generalization of the random walk which depends
on the previous position of a particle [1]}, [2]}.
This model can be converted to a walk model in which the walker has a two-dimensional internal state
\(\mathbb {C}^2\)  whose bases are represented by \(|L\rangle =[1,0]^\top \)  and \(|R\rangle =[0,1]^\top \) ,
and the moving weights to the left and right neighbors
are given by two-dimensional matrices \(P\)  and \(Q\)  so that \(P+Q\)  is a transition matrix which preserves the \(\ell ^1\)  norm.
A quantum walk [3]}, [4]}, [5]} is known as a quantum analogue of the correlated random walk in that
the matrix \(P+Q\)  is changed to a unitary matrix which preserves the \(\ell ^2\) -norm.
In our previous work [6]}, we introduced
an analogue of the quantum walk over the max-plus algebra, which has a close relationship with the ultradiscretization,
and found that the Frobenius norm on the max-plus algebra is a conservation quantity.
